Statement

A statement is the assertion behind a declarative sentence, being either true or false. Statements can be written in any language.

Here are some statements:

"The sky is blue."
"All rubber duckies are yellow."
"There exists a positive integer less than its square."

Now, some examples of non-statements:

"How's it going?"
"BOOYAH!!!"

Statements can be modified with quantifiers, which alter the scope of the claim.

Contents

Universal statements

A statement with a universal quantifier is a universal statement. They involve claims about all elements in a set, that hold over an entire domain. Most theorems are universal statements.

"All rubber duckies are yellow."

Existential statements

A statement with an existential quantifier is an existential statement. They claim that an element with certain properties exists in the domain.

"There exists a positive integer less than its square."

Disproving existential statements

Disproving an existential statement is the same process as proving a universal statement. By De Morgan's laws, the negation of the above existential statement is a universal statement:

"All positive integers are greater than or equal to their square."

So, to disprove that there exists a positive integer less than its square, you will need to prove that all positive integers are greater than or equal to their square through universal generalization.

Logic & Proofs
Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form
Set Theory
Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof
Functions
Function • Floor function • Ceiling function • Inverse function
Algorithms
Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm
Relations
Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class
Number Theory
Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm
Induction
Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition
Combinatorics
Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle
Graph Theory
Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm
Recursion
Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort